The Use of Monetary Aggregate to Target Nominal GDP

NBER WORKING PAPER SERIES

THE USE OF MONETARY AGGREGATE TO TARGET

NOMINAL GDP

Martin Feldstein James H. Stock

Working Paper No. 4304

NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue

Cambridge, MA 02138 March 1993

This paper was prepared for the NBER Conference on Monetaiy Policy, January 22 and 23, 1992, and will be published in the conference volume edited by N. Gregory Manldw. The authors thank Ben Friedman, Greg Mankiw, Ben McCallum, Steve McNees, John Taylor, Mark Watson, and an anonymous referee for helpful conversations and suggestions. We thank Graham Elliott for research assistance. This paper is part of NBER's research programs in Economic

NBER Working Paper #4304 March 1993

THE USE OF MONETARY AGGREGATE TO TARGET

NOMINAL GDP ABSTRACT

This paper studies the possibility of using the broad monetary aggregate M2 to target the quarterly rate of growth of nominal GDP. Our findings indicate that the Federal Reserve could probably guide M2 in a way that reduces not only the long-term average rate of inflation but also the variance of the annual rate of growth of nominal GDP. An optimal M2 rule, derived from a simple VAR, reduces the mean ten-year standard deviation of annual GDP growth by over 20 percent. Although there is uncertainty about this value because of both parameter uncertainty and stochastic shocks to the economy, we estimate that the probability that the annual variance would be reduced over a ten year period exceeds 85 percent.

A much simpler policy based on a single equation linking M2 and GDP is shown to be almost as successful in reducing this annual GDP variance. Additional statistical tests indicate that M2 is a useful predictor of nominal GDP. Moreover, a battery of recently developed tests for parameter stability fails to reject the hypothesis that the M2 - GDPlink is stable, but the MI -GDP and monetary base - GDPrelations are found to be highly unstable. This evidence contradicts those who have argued that the M2 -GDPrelation is so unstable in the short run that it cannot be used to reduce the variance of nominal GDP growth.

Martin Feldstein James H. Stock

NBER Kennedy School of Government

This paper examines the feasibility of using a monetary aggregate to influence the path of nominal GDP with the ultimate goal of reducing the average rate of inflation and the instability of real output. We measure the strength and stability of the link between the broad monetary aggregate (M2) and nominal GD? and we assess the likelihood that an active rule for modifying M2 growth from quarter to quarter would reduce the volatility of nominal GD? growth.

Our general conclusion is that the relation between M2 and nominal GD? is sufficiently strong and stable to warrant a further investigation into using M2 to influence nominal GD? in

a predictable way. The correlation between nominal GD? and past values of M2 is, of course, relatively weak, so the ability to control nominal GD? is far from perfect. Nevertheless, the evidence suggests that a simple rule for varying M2 in response to observed changes in nominal GD? would reduce the volatility of nominal GD? relative to both the historic record and the likely effect of a passive constant-money-growth-rate rule. Our calculations indicate that the probability that this simple rule reduces the variance of annual nominal GD? growth over a typical decade is 85%.

The paper begins in section 1 with a discussion of the goals of monetary policy and of the specific form in which we shall assess the success of alternative monetary rules. Section 2 presents several alternative monetary policy rules that will be evaluated in the paper. Section 3

then discusses three issues that must be resolved if a monetary aggregate is to be useful

for

targeting nominal GD?. These include not only the strength and stability of the link between nominal GD? and M2 but also the apparent inability of the Federal Reserve to control M2in the short-term and the risk that a more explicit use of a monetary aggregate to target nominal GD? would weaken the statistical relationship that we have found in the historic evidence (i.e, the so-called 'Goodhardt's Law problem

In section 4 we present evidence about the strength of the link between M2 and nominal GD? and discuss Granger causality tests for the entire sample and for subsamples. Section 5

focus on M2 reflects a belief that s broad monetary aggregate is likely to have a stronger and more stable relation with nominal GD? than a narrower aggregate. We teat thia assumption in section 6 by examining the strength and stability of the link from the monetary base and Ml to nominal GD?, and find strong evidence of instability in both the base/GD? and MI/GD? relations. There is a large literature on the link from financial variablea to output (recent contributions include Bernanke and Blinder (1992) and Friedman and Kuttner (1992)) and our results on the apparent usefulness and stability of the M2/GD? relation are at odds with some of it. As we explain, this is due to our focus on nominal rather than real output, to particulars of specification (we explicitly adopt an error-correction framework), and to our use of recently developed econometric tests for parameter stability.

Sections 7 and 8 then derive an optimal rule for targeting nominal GD? in a simple model and compare its performance with simpler alternative rules. Although a considerable amount has been written on the theory of nominal GD? targeting, fewer studies have examined the practical aspects of nominal GD? targeting; notable exceptions are Taylor (1985), MeCsllum

(1988, 1990), Pecchenino and Raache (1990), Judd and Motley (1991, 1992), and Hess, Small and

Brayton (1992), The investigation in sections 7 and 8 is in the spirit of these studies, except that we focus on probabilistic statements about the size and likelihood of improvements resulting from using M2 to target nominal GD?. Section 9 examines the predictive validity of our M2-bssed time series models by comparing them with private forecasts. Section 10 then returns to the question of the Federal Reserve's apparent inability to control the M2 _money stock snd discusses how that problem could be remedied by broader reserve requirements with interest paid on those reserves.

1. The Goals of Monetary Policy

The widely agreed goals of monetary policy are a low rate of inflation (price stability) and a small gap between actual real GD? and potential real GD?. There is general agreement that a

low long-term rate of inflation can be achieved by limiting the rate of growth of a broad monetary aggregate sufficiently over a long enough period of time.

The monetary policy rules that we consider in this paper are all compatible with achieving any particular long-run average rate of inflation. Moreover, in the models that we consider, the short-term monetary policy rule that is selected does not affect the ability to achieve a low long-term average level of inflation. Technically, we are assuming that the Federal Reserve could set the long-run inflation rate by the identity that mean inflation equals mean money growth plus mean velocity growth less mean real output growth. Empirical evidence suggests that the long-run mean of the growth of M2 velocity is zero (a consequence of the long-run money demand functions reported in section 4 Although there is much interesting research on the relation between long-term real output and long-term money growth (a recent empirical contribution is King and Watson (1992)), the problem of setting the means is separate from the problem of term stabilization considered here. In this sense, any gains achieved by short-run stabilization are gains in addition to those achieved by choosing the average money growth

rate which achieves low long-run inflation.

The general goal of reducing the gap between actual and potential GD? in the short and medium term can be made more precise in a variety of ways. This paper takes the approach of evaluating economic performance by the variance of the quarterly nominal GDP growth rate. This focus on the variance of nominal GD? implies giving equal weights to short-term variations of inflation and of real output. Alternative measures of short-term performance that might instead be used include the variance of real GD? growth and the mean shortfall of real GD? from potential GDP. Although such measures would ignore the short-term variation in inflation rates, the desired low long-run average rate of inflation would be assured by setting the appropriately low mean growth rate of the monetary aggregate.

Judging performance by the variance of the nominal GD? growth rate is equivalent to targeting the growth rate of nominal GD? rather than a path of nominal GD? levels. Although this distinction has no implication for the long-term inflation rste, it does affect the optimal response of policy to short-term shocks to the economy. In particular, the implicit desired future path of nominal GD? is always independent of the starting point.

This can be seen more clearly by contrasting the target of minimizing the variance of the nominal GD? growth rate (around its mean for the entire sample) with the alternative target of minimizing the variance of nominal GD? around a trend with an exponential rate of growth equal to the sum of the desired rats of inflation and the mean real GD? growth rate in the sample. If the economy starts on the trend line, the two criteria are the same for the first period. But any departure from the trend during the first period implies a different ttandard for the second period The criterion of minimizing the variance of the nominal GDP growth rate ignores any 'base drift" in nominal GD?. It can be thougth of as minimizing the variance around the trend line with the starting point of the trend rebated in each period to the actual level achieved in the previous period.

Which of the two approaches it preferable depends on the types of shocks that are most likely to be encountered, the differential effects of money on real output and inflation, and the ultimate objective of monetary policy. For example, if in the extreme real output is a random walk and unaffected by monetary policy then a nominal GDP level target will result in the price level being a random walk, so that the future price level will deviate arbitrarily far from its desired fixed level. On the other hand, minimizing quarterly fluctuations in the growth of nominal GD? will result in constant (say, zero) inflation and the future price level is stabilized. Similarly, if the growth rate of potential real GD? varies significantly from quarter to _quarter, minimizing the variance of the growth rate would be the better policy. The alternative of minimizing the variance from a prespecified nominal GD? path would require a contractionary policy after a positive productivity shock even though there had been no increase in inflation and an expansionary policy after a negative productivity thock even though there had beenno decrease in inflation. We have not explored this issue in the current research.

Our tests of the strength and stability of the link between M2 and nominal GDP are however relevant whether the criterion by which policy is judged is the variance of nominal GDP around its mean or the deviations of nominal GDP from a predetermined target path. The

choice of criterion determines how the money stock should vary from quarter to quarter to minimize the relevant variance.

L Alternative Approaches to Monetary Policy

Although the Federal Reserve is concerned with inflation and real economic activity, monetary policy must be made by adjusting some monetary variable — amonetary aggregate, an interest rate or the exchange rate. In this section we discuss three possible approaches. This is far from an exhaustive set of alternatives, but rather provides a context for comparing an M2 approach to nominal GDP targeting to other commonly discussed options.

2.1 The Status Oup: Judemental Eclecticism

In practice, the Federal Reserve controls the volume of bank reserves (a monetary aggregate) by open market sales of Treasury securities. In recent years, the volume of such sales has been adjusted to target the value of the Federal funds interest rate. Thus, for time intervals up to several weeks, any disturbance in the statistical relation between the Federal funds rate and

bank reserves (ic, in the banking system's bivariate demand function for reserves) induces the Federal Reserve to alter reserves in order to maintain the desired level of the Federal funds rate. In this context, the interest rate is the exogenous variable and the volume of reserves is endogenous. For longer periods of time, the relationship is more ambiguous because the Federal Reserve's Open Market Committee (FOMC) may revise the Fed funds rate target in part in response to the magnitude of reserve growth and the corresponding movement of the narrow monetary aggregate Ml (as well as to other aspects of economic and financial performance).

It is significant that the FOMC now makes decisions and issues operating instructions to the New York Federal Reserve Bank in terms of the Federal funds interest rate and not in terms of

M2 or some other mooetary aggregate. Each of the individual members of the FOMC may vote to increase or decrease the Fed funds rate for his or her own reasons. Some members see a reduction of the Federal funds rate as a way of increasing the rate of growth of M2 and therefore of subsequent nominal sod real GD?. Others may ignore the potential impact on the money stock and choose an interest rate change because of what they regard to be the likely effect on inflation and real output.1 At times, some FOMC members may consider the effect of changes in the Fed funds rate on the international value of the dollar. Still others may emphasize the psychological effect of changes in interest rates as an indication of the Fed's resolve to fight inflation or stimulate economic activity.

We do not try to model and test an explicit interest rate rule for monetary policy or any other complex judgmental rule. Rather we take the historic record of economic performance as indicative of what the Federal Reserve can achieve by such an eclectic judgmental policy. Technically many of the statistics we report, in particular the regression R1s and tests for predictive content in sections 4 and 6 and the performance measures in sections 7 and 8, should be interpreted as providing evidence on the ability of alternative policies to improve upon past performance. Indeed, were past performance optimal in the sense that money had been used to minimize the variance of quarterly nominal GD?, then we would expect to find no historical correlation between money and future GD? growth. In contrast, were the historical M2/GD? relationship strong and stable, this would open the door to an investigation of whether this link could be exploited to control GD? more effectively than has been done historically.

2.2 ?assive Monetary ?olicy: A Constant Growth Rate of M2

A natural starting place among explicit quantitative monetary rules is Milton Friedman's proposal for a policy of constant growth of the money supply. Setting the constant growth rate of money equal to the expected growth of potential GD? minus the expected rate of increase of velocity implies a zero expected rate of inflation. Small errors in the estimated rate of growth of either potential GD? or velocity causes correspondingly small departures of inflation from price stability.

Friedman argues that a constant rate of money growth is actually likely to result in a more stable path of nominal GDP than a more active monetary policy aimed at achieving such stability (Friedman (1953)). Friedman's argument can be summarized easily in the framework in which stability is defined as the variance of the growth rate of nominal GDP. Suppose that nominal GD? growth consists of two parts, one which would be achieved under a constant growth rule and one which reflects the impact of an activist rule. Then the variance of nominal GD? growth is the sum of the variances of these components, plus their covariance. Friedman's point is that activist policy reduces volatility only if the covariance is sufficiently negative to offset the additional variance contribution from activist control.

This decomposition provides a useful way to interpret the regression results elsewhere in the literature and in section 4. If M2 enters significantly, then necessary an optimal or nearly optimal policy can reduce total volatility. However, if the regression R2 is small, then the gains from such control will be modest. Moreover, following the 'wrong" policy can increase rather than decrease output volatility.

13 Active Tareetine Rules for Monetary Policy

McCallum (1988, 1990), Taylor (1985) and others have developed and simulated alternative rules for managing monetary policy with the aim of stabilizing nominal GD? growth. We build on this literature in sections 7 and 8 of this paper by proposing an optimal rule for using monetary policy to target nominal GD? and a simple, partial-adjustment rule that approximates

the effect of the optimal rule.

As part of our analysis of these rules, we calculate the probability that they would reduce the variance of nominal GD? growth. The specific calculation we perform addresses the following thought experiment: suppose the Federal Reserve were to adopt a particular nominal GD? targeting rule and use it for a decade. Based on the data available to us from 1959 to 1992, what is the probability that the variance of quarterly nominal GDP growth would be less over this ten-year span than it would be under the status quo? What is the expected percent

reduction in the ten-year standard deviation of quarterly GD? growth under the rule and, more generally, what does the distribution of potential reductions look like? Our statistics answer these questions, and also quantify the distribution of ten-year variance reductions in two- and four-quarter growth of GD?. This calculation incorporates both the parameter uncertainty arising from working with a finite historical data set and the additional uncertainty introduced by different possible ten-year paths of future shocks to the economy. When the policy rule is designed to minimize quarterly GD? volatility, we refer to the performance measure applied to GD? as a performance bound, since by construction the monetary policy is designed to minimize the population (multiple decade, long data set) value of this ratio. Our calculations show that in principle the optimal M2 rule would have outperformed status quo policy with a rather high probability.

The complexity of the optimal rule for varying M2, even in the simple model that we analyze, suggests that explicit optimization is more relevant as a benchmark than as an actual prescription for application by the Federal Reserve. We therefore examine simpler partial-adjustment rules, which are in the spirit of the rules examined by Taylor (1985) and McCallum (1988, 1990). In particular, the rule for which we tabulate results adjusts M2 40% toward closing the gap between realized and desired nominal GD? growth. Performance measures for this simplified rule show that it would have resulted in nominal GDP stabilization close to that of the optimal rule and better than the implicit status quo policy. Moreover, long-run mean inflation would be reduced by choosing a lower mean money growth rate. Thus this rule could result in both lower mean inflation and reduced volatility of GDP growth, relative to the status quo.

3. The Usefulnessof a MonetaryTargeting Rule: Three issues

The research in this paper shows that an active monetary rule of the type described in Section 23 and studied in Sections 7 and 8 can in principle achieve a more satisfactory economic performance (as measured by the rate of inflation and the stability of nominal GDP growth) than has been achieved by the "eclectic judgmentalism" currently practiced by the Federal Reserve or than would be achieved by the passive policy of constant M2 growth proposed by Milton Friedman. We show also that the professional forecasters do not appear to have an advantage relative to a simple M2-based VAR model at forecasting nominal GD? and

therefore conclude tentatively that monetary activism based on professional forecasts may be no more satisfactory than policies based on simpler forecasting models.

The conclusion that a monetary rule can "in principle" be useful reflects our finding of a sufficiently stable link between money and nominal GD?. Two other issues must be resolved favorably in order to conclude that monetary targeting would be useful in practice as well as in principle. Briefly, the three requirements for the usefulness of a monetary targeting rule can be characterized as: (1) a sufficiently stable link between money and nominal GD?; (2) satisfactory behavior of the Federal Reserve; and (3) a limited system response to the change in monetary policy.

3.1 A Stable Link Between Money and Nominal GD?

The statistical tests presented in section 4 and 5 show that M2 has predictive content for nominal GD? and that the relationship appears to have been stable over time. More precisely, section 4 shows that the link between money and nominal GD? exists for the entire thirty-year sample. It is strong enough so that Milton Friedman's case against active policy cannot be based on the absence of an adequate link between short-run variations of M2 and nominal GD?. The evidence in Section 5 suggests that the parameters have been stable in the sense that we cannot reject the null hypothesis of parameter constancy using several recently proposed tests for parameter stability.

32 Satisfactory Behavior of the Federal Reserve

Milton Friedman and others base their argument against an activist monetary policy in part on the claim that there is an inherent inflationary bias in central bank behavioE even if the Federal Reserve could control M2 completely and knew an optimizing rule for setting M2, they would violate that rule because of political pressures or other reasons.

There is of course no way to answer that criticism fully. We do note however that the Federal Reserve and other central banks around the world have over the past decade been pursuing relatively tough anti-inflationary policies and that those central banks with greater independence have pursued that goal more aggressively. That is no guarantee about the future behavior of the Federal Reserve. Those who believe that any central bank that has discretion will eventually act incompetently or perversely may or may not be right, but they Cannot be persuaded by evidence.

Nevertheless, if our evidence on the predictive link between money and nominal GDP is accepted, those who would still advocate a passive fixed-money-growth rule would have to argue that the gain in terms of reduced inflation that results from such a policy outweighs the potential benefit in terms of output stability that can be achieved by an active rule-based monetary policy.

It seems likely, moreover, that any policy based on an explicit quantitative rule is less subject to political and other pressures than the purely judgmental approach currently pursued by the Federal Reserve. Perhaps it would be a useful further discipline if the Federal Reserve were to state the rule publicly and to explain to the financial and policy community whenever monetary policy did not conform to the rule over a period of, for example, six months, just as the Federal Reserve now announces a target range for money growth and must explain to Congress whenever it fails to achieve money growth in that range.

In addition to the question of the Federal Reserve's willingness to use a monetary rule to target nominal GDP, there is also a more technical aspect about the Federal Reserve's ability to

act in compliance with a rule that requires managing quarterly changes in M2. Recent experience shows that conventional short-run money demand equations have broken down (Feinman and Porter (1992) Evidently the Fed has been unable to estimate the volume of open market operations needed to achieve its desired changes in M2. For example, the increase of M2 at a rate of only 2.2 percent from the fourth quarter of 1991 to the fourth quarter of 1992 was below the lower end of the Fed's target range (2.5 percent to 63 percent) at a time when most Fed officials acknowledged that faster M2 growth would have been desirable. We return to this problem in section 10 and explain that the Federal Reserve could control M2 by expanding reserve requirements to include all of the components of M2. Until then, we will ignore the difference between controlling reserves and controlling M2 and will assume that the Federal Reserve can control the growth of money from quarter to quarter.

3.3 A Limited System Response to the Chance in Monetary Policy

Even if the relation between money and nominal GDP has been stable in the past, an attempt to exploit that relation in an optimizing mode could cause a change in these reduced

form parameters. Continuing to assume the old parameter values would lead to suboptimat results that could, in principle, be worse than those implied by the existing judgmental policies.

There are two sources of this possible instability. First, as discussed in section 10, to control M2 effectively would entail placing reserve requirements on its components. To the extent that this changes the M2/nominal GDP relation, the historical correlations upon which our analysis is based would become less useful. While this effect might take some time to detect, in principle these relations could be updated using new data and the policy rule modified to account for the effect of consistent reserve requirements.

The second source is more problematic, and concerns the empirical relevance of the Lucas critique of all policy analysis. One extreme form of this concern (suggested in a British context by Charles Goodhardt and known as "Goodhardt's Law") is that trying to use M2 (or any other aggregate) to target nominal GDP would break the causal link with nominal GDP and make

controlling M2 irrelevant Because we use an explicitly reduced-form model, our calculations are an obvious target for this critique. However, all extant empirical macro models are approximations —thereis no compelling reason to think that any empirical macroeconomic model incorporates the 'deep parameters' stable to policy interventions —sothis criticism is equally applicable to all exercises in this area, The empirical relevance of the Lucas critique has been the topic of considerable debate (see for example Sims (1982, 1986)), and we have little to add on this topic. Yet, we note that the tests of sections 5 and 6 suggest that the M2/GDP relation — unlike the Mi/GOP relation, the monetary base/GOP, and the relation between various interest rates and output —hasbeen stable over the past thirty years, a period which has experienced several shifts in Fed operating procedures. More generally, the research of Friedman and Schwartz (1963) that originally established the existence of a link between money and nominal GOP covered a much longer period of time with even more substantial changes in monetary policy and economic institutions. This gives reason to hope that further changes to monetary policy would have limited effects on this relationship. These concerns do, however, imply that the relation between nominal GOP and M2 should be closely monitored were the Fed to change its the approach to monetary policy.

4. Strength of the Link from M2 to Nominal GDP

The question taken up in this section is whether M2 has predictive content for future nominal GDP growth. We address this by considering quarterly historical time series data on

money, output, interest rates and prices over the period 19591 - 19922. (Data sources and transformations are detailed in Appendix A.) Visual inspection of the time series data from

19591 -_{19922,portrayed in figure 1, indicates a link between the four-quarter growth in M2}

and nominal GOP over the business cycle and indeed over longer periods. However, there appears to be less correlation between M2 and either inflation or real GOP growth.

Econometric evidence on the predictive content of various monetary aggregates for nominal GOP is presented in table 1. Each row of the table corresponds to a regression of nominal

GDP growth on a constant and three lags of the indicated variable. As discussed in Appendix A, in these regressions nominal GD?, real GD?, the GD? deflator, and M2 appear in growth rates; individual interest rates appear in first differences; and spreads appear in levels. The first numeric column nf table 1 provides the of the regression of the quarterly growth of nominal GD? against the first through fourth lag of the indicated regressors. The second and third columns report the 's from regressions of two- and four-quarter growth (current quarter growth plus growth over the next, or the next three, quarters), respectively, against the same set of regressors. The final columns report the results of F-tests for predictive content (Granger causality tests) for M2 and other financial variables entering the regressions.

The results in table 1 suggest that, over the 1959-1992 sample, there has been a systematic relationship between M2 and nominal GD?: M2 is a statistically significant predictor of nominal GD? growth at the 1% level in those regressions which include M2 or M2 in conjunction with inflation and interest rates. M2 is capable of predicting a statistically significant yet quantitatively modest amount of the movements in output at the one-quarter horizon; for example, the regressions in rows 7 and 8 indicate that M2 improves the

one-quarter ,

relativeto using lagged real GD? growth and lagged GD? inflation, by 0.127. However, at the four-quarter horizon the improvement from using M2 is more substantial, increasing the of that regression from .092 to 326. In contrast, while the regressions with interest rates alone (equation 9 and 10) have comparable if somewhat smaller R2's at the one-quarter horizon, their R2's at the four-one-quarter horizon are less than .18.

A conventional question in the literature on the money-output relationship is whether the inclusion of interest rates eliminates the predictive content of M2 (e.g. Sims (1972, 1980)), For the current purposes, if true this would suggest that interest rates would make a more appropriate control variable than Mi The results in table 1 indicate that, for nominal GD?, this is not the case. For example, when the 90-day T-bill rate or the Fed funds rate is added to the regression in row 8, M2 remains statistically significant; in fact, the for the four-quarter regression declines because of the inclusion of these additional interest rates which evidently have no additional predictive content at this horizon.

The specifications discussed so far only incorporate short-run relationships, in the sense that they relate growth rates to growth rates or changes. However, there is substantial evidence that there is a long-run relationship between the levels of money and output (both in logs) and interest rates, which can be thought of as a long-run money demand relation, Unit root tests suggest that velocity and interest rates can be treated as being integrated of order one, and cointegration tests suggest that these two variables are cointegrated (see for example Hafer and Jansen (1991), Hoffman and Rasche (1991), and Stock and Watson (1989a)), thus long-run money demand can be thought of as a cointegrating relation among these vectors. If so, then a candidate for inclusion in these output regressions is the "error correction" term which is the residual from the long-run money demand relation. Previous investigationa suggest that a unit income elasticity is appropriate (see Stock and Watson (1989a) for results and a discussion), so the money demand cointegrating vector is specified here is ZMDt = _ln(Xt/M)

-flrRt,where X

is log nominal GDP, Mt is log nominal money, and Rt is the level of the interest rate, here taken to be the 90-day Treasury bill rate. The interest semi-elasticity of money demand, fl was estimated by asymptotic maximum likelihood using the Phillips-Loretan (1992)/Saikkonnen (1991)/Stock-Watson (1989a) procedure, and one lag of the resulting estimate of ZMDt was

entered as an additional regressor in the specifications in table Thus these regressions correspond to a single-equation error correction model (see for example Hendry and Ericsson (1991)). Although this motivation for including ZMD stems from the theory of cointegration, this term has a natural interpretation in a regression of nominal output growth on money: it controls for deviations in velocity from its long-run value as determined by the interest rate.

The results in table 1 indicate that the long-run money demand residual has noticeable predictive power, for example, adding ZMD to regression 11 improves the one-quarter by .063andimproves the four-quarter i? by .078. When the money demand residual is included

in the regression, the hypothesis that money does not enter implies that the lagged first differences andthemoney demand residual do not enter; thus in the regressions with ZMD the Granger causality tests for M2 in table 1 test both sets of exclusions (on all lags of M2 growth

and on lagged ZMD The hypothesis that M2 is statistically insignificant in the one-quarter horizon continues to be rejected in these regressions.

Despite this statistical significance for M2 in these regressions, it should be emphasized that the 's for these regressions are all rather low. For example, an for a four-quarter horizon of 39% (equation 13) indicates that the ratio of the RMSE from using this regression, relative to using a constant forecast, is only .7& Looking ahead to the question of whether M2 can be used to further reduce the fluctuations in GDP, this inherent relative unpredictability of

nominal GDP growth over the past three decades places a limit on any gains from modifying the control of M2 relative to the Fed's historical behavior.

Most of the recent research has focused on the relation between money growth and real, rather than nominal, output (e.g. Bernanke and Blinder (1992), Friedman and Kuttner (1989, 1992a), Stock and Watson (1989b)). As a basis of comparison, we therefore present econometric evidence on the predictive content of M2 for real GDP growth in table 2 In the case of real GDP growth, money has substantial predictive content and continues to enter each of the

regressions at the 1% level.

It is interesting to note that M2 is significant even in the regression with the commercial paper-Treasury bill spread. Other authors, in particular Friedman and Kuttner (1992a, 1992b) (also see Bernanke (1992)), have found that the inclusion of this spread in similar regressions

has eliminated the predictive content of money. The main difference between those results and the results in table 2 is that the F-tests in table 2 include the lagged money demand

cointegrating residual, as well as lags of money growth; the F-statistic on the three lags of money growth alone in the table 2 regression with the paper-bill spread is 1.68 which, with a p-value of .175, is not significant at the 10% level. However, the t-statistic on the

cointegrating residual in this regression is 323, and the joint F-test is significant. This

phenomenon is present in the corresponding nominal GDP regression with the paper-bill spread, in which the F-test on the lags of money alone is 1.76 (p-value .16) and the t-statistic on ZMD is 3.71 In all other regressions in table 1, however, the F-test on just the lags of M2 growth is

significant at the 5% level.3 This statistical significance of the money demand residual agrees with recent independent results obtained by Konishi, Ramey and Granger (1992), who find that the logarithm of M2 velocity is a significant predictor of real GNP growth; however, Konishi, Ramey and Granger use M2 velocity and thus impose a long-run interest semielasticity of money demand of zero rather than estimating it as we do here.

The generally low predictive content of interest rates for nominal GD? contrasts with the findings for real GD?. For example, the regression of real output growth on lags of NGDP, PGDP, R-90, and the GlO_Gi (the Treasury yield spread) has a four-quarter of .384,

while its four-quarter for nominal GDP is only .192. This is consistent with previous results in the literature that emphasize the value of the slope of the yield term curve as a forecaster of real output (Estrella and Hardouvelis (1991), Stock and Watson (1989c, 1990)).

5. Stability of the link from M2 to Nominal GDP

This section examines the stability of the direct link from M2 to nominal GD?. In their investigation of the M2/output relation Friedman and Kuttner (1992a) concluded that much of the full-sample predictive content of money for both nominal and real income was attributable to the 1960's, a finding which they attributed to disintermediation during the 1970's and 1980's. As a starting point, we therefore consider whether the main findings of Section 4 are robust to using the shorter sample with Friedman and Kuttner's (1992a) starting date of 197&3.

Table 3 presents the summary statistics of table 1, evaluated over the more recent sample. In general, M2 has somewhat less predictive content in the later sample, although the deterioration in forecasting performance is modest. For example, the four-quarter for the regression with lagged nominal GD? growth and lagged M2 growth is .30 in the full sample and is .25 in the later sample. The Granger causality test statistics indicate that M2 continues to be significant, albeit only at the 5% level in most regressions rather than at the 1% level found in table 1. Because this sample period is only two-thirds the length of the full sample, one would

not expect to find as strong statistical significance of the monetary variables as over the full sample even if the relationship is stable. For this reason, a more useful statistic is the marginal

from adding money to the regressions. While the increases remain economically significant, they drop in the later sample: at the four-quarter horizon, in the regression with nominal GDP, inflation, and the 90-day Treasury bill rate, over the full sample M2 alone has a marginal of .149 and, in conjunction with ZMD, of 227; over the later subsample, these marginal are, respectively, .073 and .185. In the later sample, when interest rates, M2, and ZMD are included, interest rates are never significant at the 5% level, while M2 and ZMD are jointly significant at the 5%levelin all regressions.

The results in table 3 contrast with the findings of Friedman and Kuttner (1992a). Although the primary focus of their investigation was real output, their table 1 presents results on forecasts of nominal GNP. One of their conclusions was that, over the 1703 - 199&.4 sample, M2 ceased to be a significant forecaster of nominal GNP. In a mechanical sense, the difference between their findings and ours is explained, in order of importance, by: (i) our inclusion of the error correction term ZMD; (ii) the choice of lag length; and (iii) the alight difference in sample periods.4 If, as argued in section 4, the cointegrated model applies, then the error correction term should be included in the regression, and because ZMD includes M2 a

test of whether M2 Granger causes output should test both lags of M2 growth and the error correction term. Concerning lag length, in the regression on GDP and M2 growth, the first lag of M2 is significant but the others, considered one at a time, are not, and a joint test of significance of the fourth lags in the regression suggests choosing the shorter specification. The effect of including the final six quarters in the sample suggest that the recent slow growth of

nominal output and M2 in the face of low and declining interest rates and a sharply inverted yield curve has tilted the results somewhat towards M2 as a predictor. While we therefore prefer the specifications in table 3, those results and Friedmsn and Kuttner's (1992a) findings suggest investigating further the question of whether the M2/nominal output relation is stable. The differences between our findings and Friedman and Kuttner's ultimately point to the

limitations of simple regression statistics, and that information of a different type is needed on the stability of this relationship.

We therefore subject these relations to a series of formal tests for parameter stability. The overall purpose of these tests is to detect parameter instability when the type of instability is unknown a-priori. If it were presumed that a break might have occurred at some known date, then the simplest test for such a break would be s Chow-type test for a shift in the parameters. However, in practice the date at which the break occurred is typically unknown a-priori and the candidate break date is based upon knowledge of the historical data. In this case, the subsequent test statistic does not have its classical sampling distribution, and the precise sampling distribution will depend on the preliminary method used to select the break date. (Christiano (1992) provides an empirical example of this point; for the associated econometric

theory, see the July 1992 special issue of the Josrnal of Bssiness and Economic Statistics on

unit root and break-point tests.) The test statistics considered here handle this difficulty by explicitly treating the break date as unknown.

Three classes of tests are considered. These tests are described in Appendix B and are briefly summarized here. Tests in the first class look for a single structural break which occurred at an unknown date during the sample. These tests sre based on the sequence of likelihood ratio statistics testing the hypothesis that the break occurred in quarter k. The most familiar of these tests is the Qusndt likelihood ratio statistic (the "QLR" ststistic), which is the maximum over k of these likelihood ratio statistics; the other two tests sre the average of the likelihood ratio statistics ("Mesn-Chow") sod sn exponential average of these proposed by Andrews and Plobcrger (1991) ('AP Exp-W")L As discussed by Andrews sod Ploberger (1991), these tests are designed to have good power properties sgaiost a single break in one or more of the regression coefficients. These tests are implemented with trimming psrsmeter A =.15_(see Appendix B). For comparison purposes, we slso report the vslue of the conventional Chow test, testing for a single brcsk occurring in 1979.3 ('Chow"). However, this date is conventional in the litersture precisely because is associated with the Fed's chsnge irs operstiog procedures

and thedoublerecessions of1979-1982. Because this break date is at least in part data-dependent, conventional critical values are inappropriate and proper p-values are not readily ascertained.

Tests in the second class are similar in spirit to the Brown-Durbin-Evans CUSUM Statistic, except that the statistics here are computed using the full-sample residuals as suggested by Ploberger and Kramer (1992a, 1992b These tests are the maximum of the squared scaled partial sum process of the residuals ('P-K max') and its average ('P-K meansq"). These tests mainly have power against breaks in the intercept in the regression in question.

Unlike the previous tests, the final class of statistics are derived to have power against continuously shifting parameters. These tests, due to Nyblom (1989), are derived as LM tests of the null of constant coefficients against the alternative that the regression coefficients follow a random walk, although they also have power against single-break alternatives. Two versions of these tests are considered: the L-all statistic tests the hypothesis that all the regression coefficients are constant against the random walk alternative, while the L-fin statistic only tests the constancy of the coefficients on the financial variables (money, interest rates, spreads, and the money demand cointegrating residual) In practice, these tests often yield different inferences. Because the various tests were derived to have power against different alternatives, when used together they can provide insights into which types of instabilities, if any, are present in these regressions.

The results of these tests are presented in table 4 for the nominal GD? forecasting regressions in table 1. In all the M2 regressions but one, the only tests which reject at the 5% levelare the Ploberger-Kramer tests (ignoring the fixed-Chow test, for which we cannot compute proper critical values because of the partly endogenous break date This suggests that the constant term in several of these regressions is unstable, but that the coefficients on the stochastic regressors do not exhibit statistically significant shifts. The only case in which another test rejects at the 5% level is for regression 5,whichincludes both the Fed funds rate and the 90-day Treasury bill rate FYGM3: the QLR test rejects with an estimated break in

803. Since neither regressions 3 nor 4 reject using this statistic, this suggest that there might be some instability in the relationship between the Fed funds-T-bill spread and nominal output This spread moves with other private-public spreads (Stock and Watson (1990)); in this light, its instability is consistent with the 10% rejection of the QLR statistic in regression 16, which includes the commercial paper-Treasury bill spread. Aside from these two regressions with the private-public yield spreads, the results suggest stable regression coefficients on the stochastic variables.5

Overall, the results of this section suggest that the predictive content of M2 (as well as other financial variables) for nominal GD? is somewhat less over the 1970-1992 subsample than over the full period. However, formal tests for parameter instability fsil to reject the hypothesis that the M2 -GD?regressions have stable coefficients over the thirty-year sample, except perhaps for a shift in the iotereept.

6. Links from other monetary aggregates to Nominal GDP

At various times, the Federal Reserve has considered employing alternative financial instruments as control variables, such as the monetary base, Ml, sod interest rates. lo this section, we examine the predictive content of these other instruments for nominal GD? growth sod the stability of these foreessting relationships.

Casual evidence suggests thst the link from other monetary sggregstes to output is less stable. The Federal Reserve is required by law to announce target ranges for monetary aggregates. In recent years, the Federal Reserve hss provided target ranges for M2 sod M3 as well ss for a broader debt aggregate, but no longer provides a target range for Ml. Federal Reserve officials argue that the payment of interest on most checking accounts (a component of Ml) has increased the substitutability between Ml accounts and the components of M2 sod therefore greatly increased the volatility of Ml velocity. Jo the first two quarter of 1992, for example, at annual rates Ml grew 13.4 percent while nominal GD? increased only 5 percent.

Annual growth rates of the monetary base and of nominal GDP, real GDP, and GDP inflation are plotted in figure 2. In figure 3, the monetary base is replaced by ML In contrast to figure 1, no clear cyclical link is evident between either the base or Ml and and nominal output.

To investigate these links more formally, we apply the Statistics described in sections 4 and 5toregressions involving base money and ML Evidence on the predictive content of base money and Ml is presented in tables 5 and6.6 The most striking feature of these results is that the predictive content of these regressions is substantially less than the corresponding regressions with M2, with four-quarter 's in the range 0.09- 020,compared with 's in table 1 of

almost 0.40. In the regressions with interest rates, the monetary base fails to be statistically significant at the 5%level,and Ml is no longer significant at the 10% level.

The stability of the base, Ml, and interest rate regressions are examined in tables 7 and 8 using the tests for parameter constancy described in section 5. Thehypothesis of parameter constancy is rejected overwhelmingly for base money, with every regression having at least one Statistic which rejects stability at the 1% level. The evidence against stability for Ml is equally strong. Interestingly all the rejections for Ml result from the break-point tests rather than from Nyblom's (1989) tests for time varying parsmeters, suggesting a regime-shift in the parameters rather than a slow evolution. In both the base and Ml regressions, the break date is estimated to be in the early 1980's, perhaps reflecting the widespread introduction of interest-bearing checkable deposits during this period. In contrast, the regressions with interest rates only in table 4 suggest that the interest rate relations are relatively stable. The instability of the base and Ml regressions provide some insight as to why the base and Ml are insignificant when interest rates are also included in the regressions: even if these variables have predictive content, the nature of that predictive content varies over time and the more stable interest rate relations "drive out" the two narrow monetary aggregates.

Several conclusions emerge from these results. Neither Ml nor the monetary base have substantial predictive content for GDP over the full 1959-1992 sample, and both aggregates are no longer significant once interest rates are included in the regressions. Moreover, the link

between these two aggregates on the one hand and nominal GD? growth on the other is unstable, with the stability tests rejecting in most specifications at the 1% level. While the link between interest rates and GDP growth appears to be more stable (with the exception of the term structure spread), the predictive content of interest rates for nominal GDP growth is substantially less than that of M2.

7. Optimal Nominal GDP Growth-rate Targeting: Performance Bounds

7.1 Methodoloav

We now turn to the task of estimating what the volatility of key economic variables would be, were the Federal Reserve to follow a nominal GD? targeting rule. Answering hypothetical questions such as this is central to the empirical analysis of macroeconomic policies. A standard approach to answering such questions, which we employ, is to adopt an empirical

macroeconomic model, to change one of its equations to reflect the policy rule in question, to solve the model with this new equation, and then to compute summary Statistics and

counterfsctual historical simulations which illustrate the effects of the change. In the context of evaluating the effect of nominal GD? targeting, this strategy was used by Taylor (1985), McCallum (1988), and Pecchenino and Rasche (1990) to evaluate various targeting rules, although the rules and/or empirical models used in these studies differed.

The empirical models we consider are a series of VAR models of the form (1), The focus here is on constructing performance bounds which measure the best outcome which the Fed could achieve were it to adopt a nominal GDP targeting strategy, relative to the performance of its historical monetary policy. As discussed in Section 3, we therefore make three admittedly extreme assumptions: that the monetary instrument in question is perfectly controllable; that the Fed could adopt the GDP targeting rule which was optimal over the 1959-1992 period; and that changing the rule by which money growth is set does not change the dynamics of the rest of the system and, in particular, does not change the relationship between money and output,

inflation, and interest rates. Completely satisfying these assumptions are unrealistic and one could not expect to achieve the performance bound in practice. Nonetheless, the computation of such a bound is a useful step were the performance bound to indicate little room for

improvement beyond historical Fed policy, there would be little reason to switch to a nominal GDP targeting regime.

To determine the optimal GD? targeting policy, we adopt the objective of minimizing the variance of GDP growth. It should be emphasized that this differs from the performance criterion used by McCallum (1988), who examined the deviation of the level of nominal GD? from a constant growth path of 3% per year. The key difference is that, in attempting to stabilize the growth rate rather than the level around a constant growth path, we are permitting

base drift in the target. As discussed in section 1, not permitting base drift has the feature —

whichto us seems undesirable —ofleading to a policy of inflating when nominal GDP is below its target path but is growing stably at 3% per year, and of tightening when GDP growth is stable at 3% but GDP is above its target path.

Because of lags in data availability, the Fed is unable to measure shocks to the economy as they occur. The money control rules considered here therefore set the money growth rate in the current quarter as a function of economic data through the previous quarter.7

The_OptimalControl Rule.

Theclass of models we work with are VAR's of the form,

(la) =

x

+

A(L)xti

+

Ay(L)Yti

+_Axm(L)mti+_5xt (ib)

Yt fly +

Ay(L)X51+

Ayy(L)Yt1

+ Ay(L)m1 + Yt

(ic) mt =

m

+_Amx(L)x51+ Amy(L)Yt.i + Amm(L)mti + mt

where x is the growth rate of nominal GD?, Y denotes additional variables, such as inflation as measured by the GD? deflator, and mt denotes the monetary variable of interest, for example

the growth rate of M2. The model dynamica are summarized by the lag polynomials A(L) and the error covariance matrix, Z —_EEtct'. To implement the optimal control algorithms we assume that the VAR is stable, that is, the roots of I-A(L)L all fall outside the unit circle. To simplify exposition we henceforth assume that variables enter as deviations from their means so that the intercepts can be omitted.

The rules considered in this paper are specified in terms of growth rates of money and output These rules automatically adjust for historical shifts in the level of velocity because target money growth rates are computed from past growth rates rather than levels. These rules do, however, assume a constant mean growth of velocity. Although M2 velocity growth has had a mean of approximately zero over the 1959 -1992period, in principle it is desirable to permit the mean growth rate of velocity to change with interest rates, and to consider rules which adjust for persistent nonzero growth in velocity. Including a levels relation between velocity and the interest rate in (1) is a natural way to do this, and the result would be a vector error correction model. The empirical results of section 4 suggest that this error correction term (the long-run money demand residual) should enter this specification. Although the general nature of the calculations for a vector error correction model are the same as for the

VAR model analyzed here, the details differ, and the analysis of the vector error correction model is beyond the scope of the investigation and ia left to future research.

Let =

't

''P',

zt

=

xt

c')', Azm(L) =

[Axm(L)AyM(L)']', and let Azz(L) be

the matrix with (1,1) block Axx(L), (1,2) block Ay(L), (2,1) block Ayx(L), and (2,2) block Then (1) can be rewritten,

(2a) =

Azz(L)Zti

+ Azm(L)mti +

(2b) mt =

Amz(L)Zti

+ Amm(L)mti +

The roots of Azz(L) are assumed to lie outside the unit circle, so that Czz(L) (I -_Azz(L)11exiata. Then (2a) can be written,

(3) =_r(L)mt+_Czz(L)czt

where r(L) =

C(L)Am(L).

Let rxm(L) denote the (1,1) element of r(L) and let Cxz(L) denote the first row of Czz(L).

The optimal control problem is to choose the money growth rule which solves,

(4) mm var(xt) =_{var[rxm(L)mti + Cxz(L)czt]}

Because m is assumed to be a function of data only through the previous quarter, the solution to this problem has the form, m =

d(L)zt

,

whered(L) solves (4). The solution sets

(5) _I'xm(L)mti+

Cz(L)czt2

where Cz(L) =

72CzL2,

so m =

rxm(L)'Cz(L)szti

and d(L)

rxm(L)1Cz(L).

The rule m =

d(L)szi

is expressed in terms of the shocks to the x equations (2a). In terms of implementation, it is more natural to express the rule in terms of actual historical data. This mathematically equivalent form of the rule is obtained by expressing EZt1 in terms of the data using (2a). The optimal control rule thus is,

(6)

where l(L) [1 +

d(L)Am(L)L['d(L)[I

Azz(L)L]. The controlled system is thus

given by (2a) and (6).

A primary measure of the performance of the optimal rule (6) considered here is the ratio of the standard deviations of the variables when the system is controlled, relative to the

standard deviation of the variables when the system is uncontrolled. To make this precise, let r• denote the ratio of the standard deviation of the i-th variable in (1) under the optimal control rule to its standard deviation in the uncontrolled case. Let F(L) denote the moving average lag polynomial matrix of the uncontrolled system, that is, F(L) (I-A(L))1, where A(L) is the matrix lag operator with elements AXX(L), etc. in (1), Let F(L) denote this matrix when the system is controlled using the optimal feedback rule (6), so that F(L) =

[(F(L)

_0)'

(F(L) 0)]', where F(L) =

_Czz(L)

+ rm(L)d(L)L and (L) d(L)L Let Z

denote the i-th variable in Z1 when the system is controlled (so that Z F(L)czt). Finally, let ci denote the i-th unit vector. Then the performance measure r is,

(7a)

r =

_{{var(ZTt)/var(Zit)}½}

(7b) =

{e'7iF

_{E5F ei/ej..lFjEEFjei}½.} EconontegricInference.

Becausethe coefficients of the VAR (1) are unknown, r must be estimated. A natural estimator of r1, f1, is obtained by substituting the empirical estimates of F(L), F(L), and E into (7b). However, in evaluating the distribution of r, two sources of uncertainty need to be addressed. The first is the conventional sampling uncertainty which arises because only estimates of the VAR parameters are available. The second source of uncertainty arises because, for any set of fixed VAR parameters, different shocks to the system will result in different realizations of Z and Z, so that the ratios of the sample variances computed using these shocks will differ from the population variances in (7a), Both sources of uncertainty need to be addressed in estimating the distribution of the performance measures. For example, one might wish to know the probability of realizing a decade-long sequence of shocks which have the perverse effect of making the optimal policy destabilizing relative to maintaining the status quo, that is, the probability of realizing r greater than one simply as a result of adverse shocks.

The statistics reported below estimate the distribution of variance reductions which would be realized over a ten year span, were the Fed to adopt the optimal policy (6). The first source

of uncertainty, parameter uncertainty, can be handled by conventional means. Because r is a continuous function of the unknown VAR parameters and because those parameters are have a joint asymptotic normal distribution, the estimator has an asymptotic normal distribution. In principle, this asymptotic distribution of can be computed using the 'delta' method, although we employ a numerically more convenient technique (discussed below),

The second source of uncertainty, shock uncertainty, can be handled by considering the distribution of the sample estimator

(8) rilA,Zh. =_{vãr(Z.JA,E, h)/vãr(Z1tIA, E)}½

where vãr(Z11A, Z)) denotes the sample variance of a realization of of length N (say) generated from the VAR (1) with parameters A and E, and where vãr(ZA, h) denotes

the corresponding sample variance when Z is generated from the controlled system (2a) and (6) with the parameters Azz(L), AZm(L), Ezz =

ZtZt"

and hZm(L). With the additional assumption that is normally distributed N(O, E), these parameters completely describe the uncontrolled system (1) and the controlled system (2a) and (6), Conditional on these parameters, the statistic (8) is a ratio of quadratic forms of normal random variables, and a variety of techniques are available for computing this conditional distribution. For example this can be computed by stochastic simulation, which is the approach used by Judd and Motley (1991) to estimate ranges of inflation and output growth produced under McCallum's (1988) monetary base rule (holding constant the model parameters and the control rule), and by Judd and Motley (1992) in their investigation of using interest rates as intermediate targets.

The measures of uncertainty reported in this and the next section combine the parameter and shock uncertainty arising from using the optimal rule (6). This was done using Monte Carlo methods. Specifically, in each Monte Carlo draw a pseudo-random realization of (A, ) was_{drawn from its joint asymptotic distribution; F(L) was computed using the submatrices} Azz(L) and Am(L), using the estimate of hmZ(L) obtained from the historical US. data;

pseudo-random realizations of length N were drawn from from atoehastic steady states of the controlled and uncontrolled system; and the sample variance (8) was computed. The distribution of these sample variances estimates the distribution of r given la(L)L8 Throughout, N =40was used, corresponding to a ten-year span.

In general the distribution of is asymmetric (1byconstruction is nonnegative but can be arbitrarily lsrge The distribution of r is therefore summarized by its mean, median, and 10%and90% percentiles. In addition, the fraction of realizations of r which would be expected to fall below one —that is, to indicate reduced volatility under the control rule —is also reported.

72 Empirical Results

The optimal control algorithm was applied to two VAR's using quarterly data over the 1959-1992 period. In botb models, the optimal rule minimizes the variance of quarterly nominal GD? growth, with M2 as the instrument The first model includes quarterly growth in GD?, quarterly inflation as measured by the GDP deflator, the quarterly growth of interest rates, and the quarterly growth of Mi This use of growth rates of interest rates, rather than their changes, differs from the specifications of sections 4 and 5. While this modification hss negligible effect on the estimated distributions of the performance measures, it prevents interest rstes from taking on negative values in the simulations used to compute the performance

measures.

Estimated performance measures and their distributions are reported in table 9 for two systems. Because the objective is to minimize the variance of nominal GD? growth, for nominal GDP growth these ratios represent performance bounds.

First consider the system in panel A. The point estimate of rGDP is .840, but the mean and median of the distribution of ten-year realizations of rGDP is somewhat larger, approximately .88. The mean ratio for four-quarter growth in GDP drops to .76. While the spread of the distribution also increases, the 90% point remains approximately constant, and the

fraction of realizations of rGDP under one is approximately 90%. In short, over a ten year span the expected effect of the optimal GD? rule would be to reduce the standard deviation of annual GDP growth by one-fourth; in nine Out of ten decade-long spans the optimal rule would result in at least some reduction in the variance of nominal GD?.

The reductions in the volatility of real GD? and GDP inflation (not shown in the table) are less than for nominal GD?. At the four-quarter horizon, the GD? targeting rule results in a mean improvement of only 6.6% for inflation and 12.6% for real GDP. However, in two-thirds of simulated decades the volatility of inflation is reduced, while in three-fourtha of the decades the volatility of real GDP growth is reduced.

The main findings from this exercise are robust to using the funds rate rather than the 90-day Treaaury bill rate as the financial variable. In this system, the optimal monetary policy still reduces nominal GD? volatility in 83% to 87% of the decades, depending on the horizon. The mean reductions for inflation volatility and real GD? volatility are again more modest than those of nominal GD?. However,the optimal policy results in reductions of the volatility of annual inflation and real output in, respectively, three-fifths and three-fourths of the simulated

decades.

73 Counterfactual Historical Simulations and Interpretation

Supposing the Fed had optimally used M2 to reduce GDP volatility, how might the economy have performed over the 1959 -1992_{period? Answering this question both is of interest in its} own right and provides a vehicle for illustrating the dynamic interactions in the model. Because the VAR captures the historical correlations between lagged money and future output, it is a useful framework for computing the performance bounds reported in the previous section. It is, however, arguably less well suited for performing counterfactual simulations, for several reasons. The model does not impose any restrictions implied by economic theory and thus is at a minimum inefficiently estimated; because structural shocks are not identified (in the sense of structural VAR analysis), simulated responses to shocks are difficult to interpret.

Nonetheless, the computation of counterfactual simulations sheds light on the dynamic properties of the model.

With these Caveats in mind, we therefore simulate the path of nominal GD? under the optimal policy rule. The simulated path is computed using the historical shocks to the first three equations in the system, with M2 determined using the ex.post optimalcontrol rule. This

simulated path, computed from the system in panel A of table 9, is plotted in figure 4 along with the actual path of GD?. The optimal policy rule would have produced markedly different paths of money and interest rates, but only somewhat different paths of nominal GD?, real GDP, and inflation, relative to the actual data.

A convenient way to summarize the optimal control rule is in terms of its impulse response function to shocks to GDP, inflation, and the interest rate; this impulse response function is d(L) given following (5) Thechange in the log of money in response to a one-standard deviation error in each of the three equations for the other system variables is plotted in figure 5. These shocks have not been orthogonalized so the impulse responses have no ready structural

interpretation. However, for a given system this impulse response facilitates the comparison of the optimal rule to the simpler rule examined in the next section.

8. Performance of Alternative M2 Growth Rules

8.1 SimolerNominal GD? Tareeting Rules

The optimal rule provides a bound by which to gauge the potential performance of alternative nominal GD? targeting schemes. As practical advice, however, the rule has some shortcomings. It involves multiple lags of several variables and thus would be rather complicated to follow. More importantly, the optimal rule depends on the specified model; because all empirical models are best thought of as approximations, as long as these approximations 'fit" (for example, forecast out-of-sample) equally well, there is no compelling reason to choose the optimal rule from any one model. Thus, it is natural to wonder whether

there are simpler money growth rules which would result in performance nearly as good as that achieved by the optimal rule, but which are simpler to explain and to implement and which do not hinge on any one model specification.

In this section we therefore consider alternative, simpler models for targeting nominal GDP. In doing so, we parallel the investigations of simple money growth rules by Taylor (1985), McCallum (1988), Hess, Small and Brayton (1992), and Judd and Motley (1991) and extend this work to the distribution of the performance measures r. The money growth rules considered here have the partial adjustment form,

(9) (mt - =

A(p - xti)

+ (lAXm51

-where is the target growth rate of nominal GDP, m is the mean money growth rate, and 0 < A <L Thus money growth adjusts by a fraction A when realized GDP growth in the previous quarter deviates from its target value by the amount

-x_1.

It was suggested in section 4 that long-run money demand is well-characterized as a cointegrating relationship between money, nominal GDP, and interest rates, with a unit income elasticity. If interest rates are 1(1) with no drift (an empirically and economically plausible specification), velocity growth has mean zero. Thus m is set to equal and the rule (9) simplifies to mt =

-Axti

+ (1-A)mi. As in section 7, the rule (9) is implemented in its deviationsfrommeans form, so that m and x are taken to be deviations from their 1960

-1992_averages.

The effect of the partial adjustment money growth rule (9) can be evaluated using the techniques of section 7.1. For example, the formula (7) for the performance measure r1 is as described in Section 7.1 except that the rule (9) replaces the optimal rule (6). Econometric inference concerning the performance measure can also be computed using the procedure described in section 7.1.

8.2. Empirical Results

The partial adjustment rule (9) was examined on a course grid of values of A between .1 and i. In general, the performance measures r were insensitive to the choice of A for 2 A .4; within this range, no value of A dominated in terms of variance reduction at all horizons. The results for A .4 are shown in table 10 for the two systems analyzed in table 9.

The striking conclusion from table 10 is that this simple partial adjustment rule produces nearly the same distributions of performance measures as does the optimal rule. The partial adjustment rule results in a somewhat lower fraction of simulated decades of improved

performance for nominal GDP at the quarterly horizon —only 70%, compared with 88% under

the optimal rule —but85% of the simulated decades have reduced annual nominal GDP volatility. As is the case under the optimal rule, under the partial adjustment rule the

improvements in inflation and real output variability are less than for nominal GDP. However, the partial adjustment rule still results in improvements in inflation and output in two-thirds of the simulated decades.

The results in panel B of table 10 indicate that these findings are robust to replacing the 90-day Treasury bill rate with the funds rate. Overall, according to these performance measures the simple rule comes close to achieving the reduction in nominal GDP volatility of the optimal rule and is robust to changing the interest rate used in the specification.

8.3 Counterfactual Historical Simulations and Interpretation

The fact that the simple rule provides a close approximation to the optimal rule suggests that the counterfactual historical values simulated using the partial adjustment rule will be close to the counterfactual values based on the optimal rule. This is in fact the case. The actual and simulated values of annual GDP growth for the system with the 90-day Treasury bill rate are plotted in Figure 6. A comparison of figures 4 and 6 reveals only slight differences between the historical values of output growth under the two rules; perhaps the largest difference is the decline in output in 1972 under the partial adjustment rule.

The impulseresponses of the partial adjustment rule are plotted in figure 7. (These impulse responsesare the lag polynomial d(L) in the representation mt d(L)szt, which is obtained by solving (2a) and (9) the plotted impulse response are scaled by the standard deviation of and so represent responses to one-standard-deviation changes in Ezt.) _Althoughthe simulatedoutput andinflation paths are quite similar under the two rules, the impulse responses of therulesare quitedifferent. Clearly the partial adjustment rule is not an approximation to the optimalrule, in thesensethat its impulse response function approximatestheimpulse response function of the optimal rule. However, its effect on_{nominal output (and}alsoon inflation and real output) is closetothat of the optimal rule. A partial explanation forthisis

that, as was emphasized in section 4, the estimates of the short-run effect of_{money on output,} while statistically significant, is still rather small so thatrather_{different money growthpaths} can have similar,modest effects onnominal output and inflation. More generally, these results indicate that the objective function of the variance of nominal GDPisratherflatwith respect to various money growth rules.9

9. Adjusting Monetary Policy to Consensus Forecasts

The empirical analysis in sections 7 and 8 uses a simple VAR model to derive and to evaluate policy rules. This analysis assumes that these low-dimensional models adequately capture stable historical correlations and that the remaining predictable structure in GDP is limited. If the VAR's have performed worse than alternative forecasting systems, then one would be reluctant to place much weight on them in designing or evaluating monetary policy. This section assesses the predictive performance of our simple VAR model by comparing it to professional economic forecasts: had our simple VAR models been run historically, would they have produced forecasts of nominal GDP as good as the historical professional record? McNees's (1986) comparison of ex-ani'e forecasts indicates that, at least for some economic variables, VAR's are capable of performing as well or better than conventional professional

forecasting models. The VAR's examined in McNees's study have more variables and a different structure than those here, however, so that work does not directly address our models.

We therefore provide evidence on how our models would have performed over this period, relative to those of private forecasters. Of cours